Magnetometry with nitrogen-vacancy ensembles in diamond based on infrared absorption in a doubly resonant optical cavity

0.54 MB·English

by  Yannick Dumeige

Magnetometry with nitrogen-vacancy ensembles in diamond based on infrared
absorption in a doubly resonant optical cavity
Y. Dumeige,1,∗ M. Chipaux,2 V. Jacques,3 F. Treussart,3 J.-F. Roch,3
T. Debuisschert,2 V. Acosta,4,5 A. Jarmola,5 K. Jensen,5 P. Kehayias,5 and D. Budker5
1UEB, Universit´e Europ´eenne de Bretagne, Universit´e de Rennes I and CNRS,
UMR 6082 FOTON, Enssat, 6 rue de Kerampont, CS 80518, 22305 Lannion cedex, France
2Thales Research and Technology, Campus Polytechnique, 91767 Palaiseau, France
3Laboratoire Aim´e Cotton, CNRS, Universit´e Paris-Sud and ENS Cachan, 91405 Orsay, France
4Hewlett-Packard Laboratories, Palo Alto, CA, USA
5Department of Physics, University of California at Berkeley, CA 94720-7300 USA
3 (Dated: December 11, 2013)
1
We propose to use an optical cavity to enhance the sensitivity of magnetometers relying on the
0
detection of the spin state of high-density nitrogen-vacancy ensembles in diamond using infrared
2
opticalabsorption. Theroleofthecavityistoobtainacontrastintheabsorption-detectedmagnetic
n resonance approaching unity at room temperature. We project an increase in the photon shot-
a noise limited sensitivity of two orders of magnitude in comparison with a single-pass approach.
J
Optical losses can limit the enhancement to one order of magnitude which could still enable room

4 temperature operation. Finally, the optical cavity also allows to use smaller pumping power when
it is designed tobe resonant at both thepump and the signal wavelength.
]
s
c PACSnumbers: 76.30.Mi,78.30.Am,07.55.Ge,42.60.Da
i
t
p
o I. INTRODUCTION improving the magnetic field sensitivity. We first recall
. the parameters which set the magnetometer sensitivity.
s
c The negatively charged nitrogen-vacancy (NV−) cen- We then theoretically investigate the extension of
i this detection scheme to the case where the diamond
s ter in diamond can be used as a solid-state mag-
−
y netic sensor due to its electron spin resonance (ESR). crystal hosting the NV ensemble is inserted inside a
h high-finesse optical cavity, as it is usually done in cavity
The center can be optically polarized and its polariza-
p ring-down spectroscopy20. Finally we determine the
tion detected through the spin-state dependence of the
[
luminescence1,2. Sensors based on a single NV− cen- improvement of the magnetometer response associated

1 ter have the potential to achieve atomic-scale spatial with the cavity quality (Q) factor.
v resolution3–5. Ontheotherhand,magneticfieldsensitiv-
8
itycanbeenhancedbyengineeringthediamondmaterial
0
inordertoincreasethe spindephasing time whichlimits
8
0 theESRlinewidth6. Themagneticresponseofanensem- II. SINGLE-PASS PHOTON SHOT-NOISE
. bleofNV− centers7–10 leadstoaluminescencemagnified LIMITED MAGNETIC FIELD SENSITIVITY
1
by the number N of the sensing spins. Such collective
0
response also improves the signal to noise ratio and the The principle of the method is similar to the one used
3
1 sensitivity by a factor √N since the quantum projection inopticalmagnetometersbasedontheprecessionofspin-
: noise associatedwith the spin-state determinationscales polarized atomic gases21. The applied magnetic field
v
as2,11 √N. value is obtained by optically measuring the Zeeman
i
X Currently, the sensitivity of practical magnetometers shiftsoftheNV− defectspinsublevelsviatheabsorption
−
r based on the detection of red luminescence of the NV monitoring of the IR probe signal. The photodynam-
a −
ensemble is limited by background fluorescence and ics of NV centers are modeled using the level structure
poor collection efficiency. Recent advances in diamond depicted in Fig 1.a). The spin sublevels m = 0 and
s
engineering have enabled improvements in collection m = 1 of the 3A ground triplet state are labeled 1
s 2
± | i
efficiency which should improve fluorescence based and 2 and separated by D = 2.87 GHz in zero mag-
sensors,12–16 but here we consider a different approach. netic|fiield. 3 and 4 are the respective spin sublevels
In addition to the well-known transitions leading to red of the 3E e|xciited le|viel. Levels 5 and 6 are single-
| i | i
fluorescence, it has been shown recently the existence state levels related to the infrared absorption transition.
of an infrared (IR) transition related to the singlet The relaxation rate from state i to j is denoted k . As
ij
states17,18. This transition can be exploited in an k k (see Table I in Appendix E), the system is
35 45
≪ −
IR-absorption scheme with an increased sensitivity as optically polarized in m = 0 while pumping the NV
s
compared to the usual scheme19. In this paper we show centers via the phonon sideband. Without microwaves
that using IR absorption detection in combination with applied, there is reduced population in the metastable
a high-finesse optical cavity, it is possible to tune the singlet state, 6 , correspondingto a minimal IR absorp-
| i
absorption contrast to order unity thereby dramatically tion signal. Under application of resonant microwaves
2
with frequency D γB/(2π), where B is the magnetic by19,22,23
± −
field projection along one of the four NV orientations
and γ = 1.761 1011 s−1T−1 is the gyromagnetic ratio, Γmw hc
× δB = , (2)
population is transfered from ms = 0 to ms = ±1 sub- γC rPStmλS
level resulting in greater population in the metastable
where P is the measured IR probe beam signal out-
singlet and lower IR signal transmission. The experi- S
put power (wavelength λ ), and t is the measurement
mental configurationfor single-passabsorptionmeasure- S m
time. Assuming no power broadening from either pump
ments is shown in Fig 1.b). The output transmission is
ormicrowaves,theESRFWHMisrelatedtotheelectron
∗
spin dephasing time by Γ = 2/T (in rad/s). For a
a) mw 2
Excited state 3E detected IR signal power PS = 300 mW using Eq. (2)
ms=±1 4〉 k45 wweithobptaarianmaetsehrovta-nluoeissegliivmeniteidn Tmaabg.neItIicoffieAldppseenndsiixtivE-
m=0 3〉
s k ity of 20 pT/√Hz in a single-pass configuration at room
35
k 5〉 temperature. Note that considering this IR signalpower
W k 42 IR : W Γ and a beam waist diameter of 2w0 = 50 µm there is
P 31 S no saturation of the IR absorption (see Appendix B).
k 6〉 Forthissingle-passconfiguration,thecontrastcannotbe
62 improvedby increasing the thickness of the sample since
Metastable
ms=±1 2〉 D W T k61 singlet level ffroormLtlharegaebrstohrapntitohnecprousmspsepcteinoentraantdioNnVd−epctehn(t≈er1d2e0nsµitmy
mw 1
m=0 1〉 ofTab. I andII)its absorptionbecomestoostrong. The
s
Ground state 3A
2 photonshot-noiselimited sensitivitycanbe comparedto
b) the spin-noise limited sensitivity
I NV I (z) I
0,P P out,P 2
δB = , (3)
q γ nVT∗t
2 m
I0,S IS(z) Iout,S where we take into accountpthrough the factor of 2 that
−
z onlyonefourthoftheNV centersareorientedalongthe
0 L magnetic field24, n is the NV−-center density and V is
the illuminated diamond volume. In the single pass con-
FIG. 1. a) Level structure of NV− center in diamond. The figuration of Ref. [19], the spin-noise limited sensitivity
photophysical parameters related to this six-level system are is about 0.02 pT/√Hz.
given in Tab. I of Appendix E. The solid (dot) lines corre-
spondtoradiative(non-radiative)transitions. D≈2.87GHz
is the zero-field splitting of the ground state. b) Diagram of
III. SENSITIVITY ENHANCEMENT
the experimental configuration used to measure the single-
passcontrast of theIRabsorption underresonant microwave
application19. I0,P and I0,S are the pump (wavelength λP) According to Eq. (2), the magnetic-field sensitivity is
and the probeinput intensities. limitedbythelowcontrast . Inparticular,atroomtem-
C
perature the contrast is an order of magnitude smaller
than at 75 K due to homogeneous broadening19. It can
measured either with or without applying the resonant
also be seen as limited by the optical depth estimated
microwaves. Thecontrast is definedasthe relativedif-
C to only 2.2 10−2 for the experimental demonstration
ferenceintheIRsignaldetectedafterpropagationinthe ×
reported in Ref. [19]. However, the optical depth can
diamond crystal of length L
be increased by using a cavity resonant at the IR signal
wavelengthresultinginanincreaseoftheopticalpathby
I (0) I (Ω )
= out,S − out,S R , (1) a factor proportional to the finesse of the cavity. More-
C Iout,S(0) over,using a diamond crystal thickness smaller than the
pump absorption length allows to overcome the issue of
where I (0) [I (Ω )] denotes the IR signal inten- thepumpdepletionandtoobtainagoodmicrowavefield
out,S out,S R
sitywithout[with]the applicationofthemicrowavefield homogeneity along the crystal. We consider the Fabry-
whoseRabiangularfrequencyisdenotedΩ . Wecanes- Perot cavity configuration depicted in Fig 2a), consist-
R
timate the photon shot-noise limited sensitivity at room ingofatwo-sidecoatedbulk-diamondplatecontaininga
temperature for an optical power compatible with the highNV−-centerdensity(largerthan4 1023m−3). We
×
IR saturation intensity. For an ESR full-width-at-half- consideranall-passFabry-Perotcavityfor the IRsignal.
maximum (FWHM) Γ , the magnetic field sensitivity This means that the amplitude reflectivity of the back
mw
(or the minimum detectable magnetic field) of a mag- mirror is ρ = 1 and of the input mirror reflectivity
back,S
netometer based on IR absorptionmeasurement is given is ρ < 1. Regarding the pump, we consider either
in,S
3
-
a) NV pendix C at the first order
ρ
back
Iout,P C λP=532nm bP fP R (0)= ε−a0 2
S
ε+a
D F Iout,SI0,S I0,P λS=1042nmρin 0 bS fS L z RS(ΩR)=(cid:18)(cid:18)εε−+aaΩ0ΩR(cid:19)R(cid:19)2. (4)
b)RS=Iout,S/I0,S cin)teMnasxitiymmala ignntrifaiccaatviiotyn pfaucmtopr The finesse ofthe cavity given in Eq. (C5) can also be
written at the first order in ε and a
100% i
102
L=L π
RS(0,λ1) mwoff 10 0 FS = ε+ai. (5)
R(0) mwon
S 1 L=10L
0 where i=0 for off-resonancemicrowavesand i=ΩR for
R(Ω) 0 λ F on-resonance microwaves.
S R S P
λ1 λ0 1 10 102
FIG. 2. a) All pass cavity (we consider a perfectly reflecting
1. Optimal cavity coupling
backside mirror |ρback,S| = 1) used for magnetic field sensi-
tivity enhancement. ρin,i is the amplitude reflectivity of the
input coated mirror. The cavity can be doubly resonant for Assumingaperfectspinpolarizationandnoadditional
the pump and the signal. C: optical circulator, F: optical optical losses, we have: a0 = 0. In this case, RS(0) =
filter rejecting the pump beam, D: optical detector. b) Re- 1 and thus the off-resonance reflected detected signal is
flectedspectrumfromthecavityforswitched-onorswitched- equalto the input signalpower P . The contrastreads
0,S
off microwaves (mw) resonant at the level |1i-|2i transition. =1 R (Ω )andthemagneticfieldsensitivityisgiven
S R
λ0 is the IR cavity resonance wavelength. c) Maximal intra- bCy −
cavity pump beam optical power magnification factor for a
givenvalueofintracavityabsorptionandtwovaluesofcavity
Γ hc
lengthsL=L0andL=10L0. Thefinesseofthecavityatthe δB = mw . (6)
pumpwavelengthisdenotedFP. L0isthecavitylengthwhich γ[1 RS(ΩR)]sP0,StmλS
−
givesacritical coupling(andthusthetheoptimalmagnifica-
tionfactor) forFP ≈100. Notethatevenwithρin,P =0,the For ε = aΩR, the incoming and outgoing fields destruc-
all-pass configuration gives a maximal magnification around tivelyinterfereattheresonantwavelengthandR (Ω )=
S R
4 dueto reflection on thebackside mirror. 0. The laser probe beam is then critically coupled25
−
to the cavity-NV ensemble system and the contrast is
equalto1. Forthis particularvaluetheoptimalsensibil-
single-pass propagation (ρ = ρ = 0) or all-pass
in,P back,P ity of the magnetometer is reached.
cavities (ρ = 1). We define the reflection of the
back,P
cavityat opticalresonanceby R =I /I with (with
i out,i 0,i
i P,S ).
2. Effects of the microwave off-resonance absorption
∈{ }
Now we consider the more realistic case of a non ideal
A. Basic principle of the cavity effect spin-polarization and material with parasitic IR losses
whichgivesa >a >0. Therearethreepossiblecases
ΩR 0
The complete analysis of the cavity has to be per-
formed numerically. In order to allow a simple inter- i) ε>√a0aΩR RS(0)>RS(ΩR)
pretationofthe results,we firstderiveanalyticalexpres-  ii) ε=√a0aΩR RS(0)=RS(ΩR) (7)
sIRio-nssigfnoarl tahbesosrepntsiiotniv.ityThaessuabmsionrgptnioonsoaftutrhaetioIRn osfigtnhael iii) ε<√a0aΩR RS(0)<RS(ΩR).
due to levels 5 and 6 and the spin polarization due Consequently, depending on the relative value of R (0)
| i | i S
to the pump beam system is simply taken into account
and R (Ω ), the expression of the contrast is different.
S R
byA thesingle-passround-tripamplitudetransmission.
S This can be taken into account by writing
We also assume a good finesse cavity at the IR signal
wavelengthand thus the input mirror reflectivity can be R (0) R (Ω )
S S R
= | − | . (8)
writtenρin,S =1−εwithε≪1. Withtheapplicationof C max[RS(ΩR),RS(0)]
theresonantmicrowavefieldwehave: A (Ω )=1 a
S R − ΩR
(a 1) whereas for an off-resonance microwave field This relation can be used to write the expression of the
ΩR ≪
we have: A (0) = 1 a (a 1). We define the op- minimum detectable magnetic field taking into account
S 0 0
− ≪
tically resonant reflectivity for respectively off- and on- the detrimental effect of the residual IR absorption due
resonancemicrowavefieldsusingthe resultsgiveninAp- to non-ideal branching ratio to the metastable state by
4
−
multiplying P by max[R (Ω ),R (0)] to obtain the intrinsically limited by NV photophysical parameters
0,S S R S
detected IR power P of Eq. (2). The fundamental ad- andby diamondintrinsic IR opticallosses. Those effects
S
vantage of the present method is that this quantity falls are quantitatively described in the next section where
under the squareroot whereas for methods basedon the numerical results are reported.
visible-fluorescence monitoring the non-ideal branching
ratio reduces the contrast by a similar amount, but
C
this quantity falls outside the square root. One can esti- B. Numerical calculations
matethatinthesameconditions,theminimaldetectable
magnetic field δB is reduced by a factor of 5 in com-
≈ The output fields Eout,i both for the pump and IR
parison with δB obtained via fluorescence method with
f signal are deduced from the input and intracavity for-
acollectionefficiencyη 0.47(seedetailsanddiscussion
≈ ward and backward propagating fields fi(z) and bi(z)
in Appendix F). The sensitivity thus reads
described Fig. 2a) using the slowly varying envelope ap-
proximation. Note that the intracavity absorption (ob-
Γ hc max[R (Ω ),R (0)]
mw S R S tained by solving the six-level rate equations) depends
δB = × .
γ|RS(0)−RS(ΩR)|s P0,StmλS nonlinearly on the intracavity intensity Ii(z) and thus a
(9) numerical optimization routine on fi(L) must be used
Inthepresentcase,therearetwocritical-couplingcondi- to deduce the reflected powers both at pump and signal
tions, thus the sensitivity δB can reach two optimal val- wavelengthsfor the targetvalues26 of I0,i (see Appendix
ues obtainedfor ε=a (solid line in Fig 2b) or ε=a . D for details on the calculation method).
Note that due to theΩfRactor max[R (Ω ),R (0)] i0n WeconsidertwoNV−centerconcentrations19,27i)con-
tahcetunaullmyerreaatcohredoffoErqv.al(u9e)s, othfeεpsmliignhimtlyumdSiffvearRleunetsfoSrfomδBthies fifigguurraattiioonn−12:: nn==42.48××11002323mm−−33anadndT2∗T2=∗ =39105n0sniis).cFoonr-
exactcritical-couplingfinesse. Thiswillbeaccuratelyde- high NV -center density, single-pass absorption is high
scribed in the numerical calculations. We first consider andthesystemislesssensitivetoparasiticopticallosses,
the case i) of Eqs. (7). Assuming ε a we have but the electron spin dephasing time is shorter than for
≫ ΩR lesslowdensitysamples. Foreachoftheseconfigurations
πΓ hc we analyze: i) the effect of the diamond crystal sample
mw
δB . (10) thickness, ii) the effect of the input power, and iii) that
≈ 4γFS(aΩR −a0)sP0,StmλS of the Q-factor of the cavity. The Q-factors are defined
by Q =2n LF /λ , (i P,S ) n =2.4 being the dia-
This means that for low cavity finesses the effect of the i d i i d
∈{ }
mondrefractiveindex andwhere we recall(see Eq. (C5)
cavityistoreducetheminimumdetectablemagneticfield
in Appendix C) that the finesse F is defined by
value by a factor equal to the finesse F π/ε. For i
S
≈
ε = √a0aΩR (case ii), the contrast is equal to zero and
π ρ A
δB reachesa singularvalueasshowninEq. (2). Finally, in,i i
F = . (12)
i
forε<√a0aΩR (caseiii),assumingε≪a0thesensitivity 1p−ρin,iAi
reads
with A the single-pass round-trip transmission. Note
i
Γ a a hc that in the case of a resonant pump field, the cavity is
δB mw ΩR 0 . (11)
≈ 4γε(aΩR −a0)sP0,StmλS dAesig=neρd in owrhdiecrhtgoivreesacthheexmacatxliymtahleinctrriaticcaavlitcyouppulminpg
P in,P
This shows that the sensitivity can be greatly impaired fieldenhancementandtheoptimalpumpenergytransfer
(i.e. δB increases) if the empty cavity finesse (π/ε) is to the NV− ensemble.
larger than that of a critically coupled cavity given by Figure3showsthemagnetic-fieldsensitivityasafunc-
π/(2a ). Moreover, Eqs. (10) and (11) show that if the tionofthecavityQ-factorQ attheIR-signalwavelength
0 S
off-andon-resonancelossvaluesa anda aretooclose, fortwocavitylengthsandthreevaluesofα whichrepre-
0 ΩR S
the sensitivity is also impaired. sents the IR-signal optical-loss due to the bulk diamond
As a conclusion, the level 6 is always partly popu- material alone. In the rate-equation approximation, the
| i
lated due to the non ideal branching ratio to the dark sensitivity reachestwo maxima(minima ofδB), the first
singlet state (k = 0). This results in absorption of correspondingtoacavitycriticallycoupledwhenthemi-
35
the IR probe beam6, even in the microwave-offstate (i.e. crowaves are switched-on and the second corresponding
no resonant microwaves applied) and the implementa- to a cavity critically coupled when the microwaves are
tion of a cavity will also increase this effect and reduce switched-off. Between these two optimal coupling con-
the detected IR photon number I . Thus, the cav- figurations, we observe a sharp decrease of the sensitiv-
out,S
ity induces simultaneously an increase in the contrast ity corresponding to a cancellation of the contrast. For
C
andareductionofthedetectedphotonnumberintheIR this particularsituation,thereflectionforthe microwave
beam. Consequently, for a given single-pass absorption, switched-on and switched-off cases are equal. The IR
the cavityfinessecannotbearbitrarilyincreasedandthe optical losses reduce the sensitivity of the cavity but for
magnetic field sensitivity δB reaches a minimum value α = 0.5 cm−1 (α = 0.1 cm−1) the best sensitivity
S S
5
23 -3 S=0.1cm-1
0.5 Hz)100 Cao) nL=f1ig00. (cid:181)1m - / nFP==243. 4/ xIP=1800MmW /m2 SS==03.c5mcm-1-1 10 CCoonnffiigg 22.. LL==1100(cid:181)0(cid:181)mm/Q/QS=S1=.28.x51x0140/4F/PI=P=3410/0IPM=4W0M/mW2/m2
/
(pT 10 0.5 z)
B 1 H
/
T
0.1 2 3 4 5 B (p 1
10 10 10 10
)
5
0. z100
H
T/ 10 Config 1. L=10(cid:181)m/QS=5.3x104/FP=160/IP=8MW/m2
p 4 2
( Config 1. L=100(cid:181)m/QS=5.1x10/FP=23/IP=80MW/m
B 1 b) L=10(cid:181)m / FP=160 / IP=8MW/m2 0.1 -3 -2 -1 0 1
0.1 10 10 10 10 10
102 103 104 105 P0,S (W)
23 -3
Config. 2 - n=28x10 m FIG. 4. Shot-noise limited magnetic-field sensitivity calcu-
0.5 Hz)100 c) L=100(cid:181)m / IP=400MW/m2 l2awte0d=fo5r0ΩµmR =var2yπin×gt1h.5eiMnpHuzt,IαRSsi=gn0a.l5pcomwe−r1.,FαoPrC=on0fiagn.d2
T/ 10 (n=28×1023 m−3) and L=100 µm we assume single-pass
p
( propagation for the pump. The cavity parameters havebeen
B 1
optimized using Fig. 3.
0.1
2 3 4 5
10 10 10 10
)
5
0. z100
H cavity finesse and thus to strongly reduce the required
/
pT 10 amount of pump power from 400 MW/m2 (single-pass
B ( 1 2 propagation) to 8 MW/m2. For n=28 1023 m−3, the
d) L=10(cid:181)m / FP=31 / IP=40MW/m ×
0.1 pump absorption is so high that for L = 100 µm a dou-
102 103 104 105 bly resonantapproachdoes notgiveanyimprovementin
the required pump power (I = 400 MW/m2). Never-
Cavity quality factor Q at 0,P
S S
theless, for short cavities (L = 10 µm) a modest-finesse
cavity for the pump (F = 31) leads to a reduction of
FIG. 3. Shot noise limited magnetic field sensitivity vs Q- P
2
factor of the cavity at the signal wavelength and for dif- the external pump power (down to I0,P = 40 MW/m ).
ferent values of IR-signal optical losses (αS). Calculations InFig. 4weplotthe magnetic fieldsensitivityas afunc-
are done for ΩR = 2π × 1.5 MHz, P0,S = 300 mW with tion of the IR signal input power P0,S for a beam-waist
2
I0,S = 150 MW/m and no optical losses for the pump diameter 2w0 = 50 µm. For thick diamond slabs, the
(αP =0). c) Config. 2 and L=100 µm, we assume a single saturationis obtained at highpower ( 10 W). For thin
passpumping. Foreachplot,thevalueofδBobtainedforlow ≥
diamond slabs, the use of high-finesse cavities reduces
Qs is about half compared to that obtained for single-pass the signalsaturationpower. Inthe highest-Q-factorcase
propagation as expected from the use of a high reflectivity
(Config. 1 and L = 10 µm), saturation starts around
backside mirror.
P 300 mW. For high signal input power thermal
0,S
≈
effects must be taken into account. Note that these ef-
fects would improve the sensitivity via the thermo-optic
can reach 0.6 pT/√Hz (0.3 pT/√Hz) corresponding to effects. More generally any nonlinear dispersive effect
almosttwoordersofmagnitudeenhancementincompar- would increase the sensitivity of the device. In this case,
ison to single-pass approaches. For strong optical losses a change in the absorption for the signal would induce a
(α = 3 cm−1) the sensitivity is still enhanced by more shiftofthecavityresonance. IntheexampleofFig. 2.b),
S
than one order of magnitude andthe performance of the if we denote λ λ the shift of the cavity, the contrast
1 0
−
cavity system is comparable with that of the same sam- would be given by [R (0,λ ) R (Ω )]/R (0,λ ) and
S 1 S R S 1
ple in a single-pass configuration at low temperature19. would have approximately the−same value than without
We now discuss the results for IR optical losses set to nonlinear effects. However the detected reflected power
α =0.5 cm−1. For n=4.4 1023 m−3, it is possible to wouldbeR (0,λ ) P andwouldbegreatlyincreased
S S 1 0,S
× ×
use a doubly resonant cavity to increase the intracavity in comparisonwith R (0) P which could reduce the
S 0,S
×
optical pump intensity and thus to reduce the required value ofthe minimumofthe detectable magnetic fieldas
externalintensityasillustratedinFig. 2c). Bydiminish- shown for example by Eq. (2).
ing the length of the cavity, the single pass attenuation Wecancheckthatalltheresultsgivenhereareconsis-
is reduced and thus it is possible to increase the pump tent with the quantum-noise limited sensitivity: i) Con-
6
fig. 1 δB = 0.2 pT/√Hz and δB = 0.06 pT/√Hz ii) tersindiamondatroomtemperature. Wefoundthatfor
q q
−
Config 2. δB =0.13 pT/√Hz and δB =0.04 pT/√Hz diamond samples with a high density of defects (NV -
q q
for L=10 µm and L=100 µm respectively. The choice center concentration larger than n 4.4 1023 m−3),
≥ ×
ofparametersforeachcaseconsideredaboveresultsfrom our configuration allows an enhancement of two orders
an optimization depending on the crystal thickness and of magnitude in comparison with single-pass configura-
NV− center concentration. Note that in the most res- tions. In the presence of high IR optical losses the en-
onant configuration (Config. 1 and L = 10 µm), the hancement is reduced to one order of magnitude. The
optimal overall Q-factor of the cavity for the probe is use of a cavity compensates for the reduction of the
around 5.3 104, giving a cavity bandwidth γ = optical depth due to homogeneous broadening at room
cav
2π 5.4GHz×muchlargerthanthe probe-laserlinewidth temperature19. Moreover,doublyresonant(forthepump
(γ × 2π 10MHz)used for single-passexperiments re- andtheprobe)cavitiescanbeusedtoreducetheamount
L
port≈ed in×Ref. [19]. For high NV− concentrations (Con- of required pump intensity (down to 8 MW/m2). Us-
fig. 2), the required Q-factor can be low ( 3 104) and ing diamond samples with a very high density of de-
thusthetotalopticalpathℓ=λ Q /(2πn≤)(×ℓ 2mm) fects (n 28 1023 m−3), this approach could be im-
S S d
≈ ≈ ×
issmalleroralmostequaltotheRayleighrangeobtained plemented using monolithic planar Fabry-Perot cavities
forawaistdiameter2w =50µm(2Z 3.8mm). Con- or integrated diamond photonic structures such as mi-
0 R
sequently, the simple planar Fabry-Pero≈tgeometry28 de- crodisk or microring resonators. For smaller defect con-
pictedinFig2a)canbeused. Finally,consideringhighly centrations (n 4.4 1023 m−3), external spherical-
≈ ×
concentrated thin samples the required Q-factor can be mirror cavities should be used.
around2 104 whichiscompatiblewithrecentmeasure-
ment repo×rted on integrated diamond microcavities29.
ACKNOWLEDGMENTS
C. External-mirror cavities
ThisworkwassupportedbytheFrance-BerkeleyFoun-
For the highest-finesse cavities, appropriate for a con-
centrationofn=4.4 1023m−3,theeffectivelengthℓis dation, the AFOSR/DARPA QuASAR program, IMOD
× andtheNATOScienceforPeaceprogram. K.J.wassup-
longerthantheRayleighrangeforthechosenbeamwaist
ported by the Danish Council for Independent Research
value (2w = 50 µm). Consequently, external spheri-
0
Natural Sciences.
cal mirrors should be used. If we consider for example
a confocal cavity, the distance between the mirrors is
L =2Z =3.8 mm. For a 100 µm (10 µm) thick dia-
cav R
mond plate, the finesse of the cavity would be F =110
S
(F =1150). Consequently, in the case of the highest fi- Appendix A: NV− six-level modeling
S
nessecavity,the Q-factorwouldbe8.4 106 correspond-
×
ingtoacavitybandwidthγ =2π 34MHzstilllarger
cav × The local density nj(z) (with j [1,6]) of the centers
than the probe-laser linewidth. We have assumed here ∈
of each levelare calculatedby solving the rate equations
distributed optical losses such as α = 0.5 cm−1; if we
S assuming dn /dt = 0. We consider spin-conserving op-
j
consider that optical losses mainly come from diamond
tical transitions. The pump excites a vibronic sideband
interface roughness, it implies that in the more unfavor-
which decays quickly via phonon emission to levels 3
able case (for the 10 µm-thick diamond plate), the root | i
and 4 . This allows us to neglect the down-transition
mean square deviation of the surface to planarity of the | i
rates due to the pump light. At z, the relation between
diamond interfaces30 has to be less than 2 nm, which is
the optical intensity and the center densities is given by
attainable with state-of-the-artfabrication techniques31.
(z) (z)= , (A1)
0
M ·N N
IV. CONCLUSION
where = (0,0,0,0,0,n)T, contains the values of
0
Theuseofacavitycanenhancethesensitivityofopti- the cenNter densities: = (nN,n ,n ,n ,n ,n )T and
1 2 3 4 5 6
− N
cal magnetometers based on IR absorption of NV cen- the matrix (z) can be written:
M
[W (z)+W ] W k 0 0 k
P mw mw 31 61
−
W [W (z)+W ] 0 k 0 k
mw P mw 42 62
 W (z) − 0 (k +k ) 0 0 0 
(z)= P − 31 35 , (A2)
M  0 WP(z) 0 (k42+k45) 0 0 
 0 0 k − k [W (z)+Γ] +W (z)
 35 45 − S S 
 1 1 1 1 1 1 
 
 
7
we assume here a closed system: 6 n = n. The Appendix C: Analytic expression of the
j=1 j
transition rates W (i = P for the pump and i = S cavity-reflectivity in the linear regime
i
P
for the IR signal) are related to the optical intensity I ,
i
the wavelength λi and the absorption cross section σi Here we consider the cavity describedin Fig. 2a) with
by Wi = σiIiλi/(hc). Assuming a low Rabi angular fre- ρback,S = 1. We denote the probe input field E0,S, the
quency ΩR, in the rate-equation approximation, the mi- reflected field Eout,S and the forward propagating field
crowave transition rate is calculated as Wmw = Ω2RT2∗/2 inside the cavity at the input mirror S(0). Introduc-
∗ F
whereT istheelectronspindephasingtime. Thecenter ingtheamplitudemirrorIRtransmissioncoefficientκ
2 in,S
density in each level is calculated by N =M−1N0. verifying κ2in,S +ρ2in,S = 1 and the round-trip phase ϕ,
we can write
(0)=jκ E +ρ A (0)ejϕ
S in,S 0,S in,S S S
F F (C1)
(Eout,S =ρin,SE0,S +jκin,SAS S(0)ejϕ.
F
Appendix B: IR-absorption cross section estimation
By eliminating (0), we can deduce the amplitude
S
F
transfer function of the cavity
In order to model the system we have to evaluate the
IR absorption (due to singlet states) cross section σS Eout,S ρin,S ASejϕ
whichhasnotbeenmeasuredsofar. Withtheaimofde- = − . (C2)
E 1 ρ A ejϕ
0,S in,S S
signing a cavity based magnetometer, the value of σS is −
important to evaluate the intracavity IR signal intensity
The intensity reflectivity of the cavity is thus given by
saturation. This completes the already reported list of
−
tphhaottoaprehysusimcamlparriozpeedrtinieTsaobf.thIegNivVeninceAntpeprsenindidxiaEm.Honedres Eout,S 2 = ρ2in,S +A2S −2ρin,SAScosϕ . (C3)
E 1+ρ2 A2 2ρ A cosϕ
we estimate σS by using the single-pass IR-absorption (cid:12)(cid:12) 0,S (cid:12)(cid:12) in,S S − in,S S
measurements described in Ref. [19]. We assume that (cid:12) (cid:12)
Atre(cid:12)sonance(cid:12)ϕ=0(2π)thereflectivityofthecavitycan
the measured magnetic field is oriented in such a way
− be written
thatthe microwavesareonlyresonantwithNV centers
of a particular orientation, i.e., one quarter of all the 2
ρ A
NV− centers24. Inthe single-passconfiguration canbe RS = in,S − S . (C4)
calculated by integrating the two differential eqCuations (cid:18)1−ρin,SAS(cid:19)
considering off-resonance pumping and a resonant exci-
In the all-pass configuration, the finesse of the cavity is
tation (including stimulated emission) for the signal
given by
π ρ A
dI in,S S
P F = . (C5)
= σ [n (z)+n (z)]+α I (z) S
P 1 2 P P 1 ρ A
dz −{ } (B1) p− in,S S
 dI
 S = σS[n6(z) n5(z)]+αS IS(z),
dz −{ − }
Appendix D: Numerical cavity-reflectivity
 calculation
where the densities n (z) with i [1,6] are the station-
i
∈
ary solutions of the rate equations corresponding to Fig. For i S,P , if and denote the forward and
i i
1a) (see Appendix A). αi with i ∈ {P,S} are the op- backward∈p{ropag}atingFfields,Bthe intracavityfield Ei can
tical losses due to light scattering or parasitic absorp- be written
tion. Calculations are carried out using the parameters
given in Ref. [19] recalled in Tab. II (see Appendix E (z)= (z)+ (z). (D1)
i i i
E). The two unknownvalues are the IR absorptioncross F B
section σS and the optical losses αP at the pump wave- With fi and bi, the slowly varying envelope amplitudes
length. The method consists in numerically finding the oftheforwardandbackwardpropagatingfieldsshownin
values of σS which gives the contrast value defined in Fig. 2a), we obtain
Eq. (1) and reported in Ref. [19]. We have then de-
ducedthatforamonochromaticexcitation(thelinewidth Ei(z)=fi(z)e−jβiz +bi(z)ejβiz, (D2)
of the IR laser is γ 2π 10 MHz γ ), the IR
L IR
absorption cross sectio≈n due×to the me≪tastable level is with βi =2πnd/λi. The field amplitudes are normalized
σ = (2.0 0.3) 10−22 m2. The uncertainties come in order to have I (z)= E (z)2. The calculation of the
S i i
± × | |
from the value of α which has been assumed to vary cavity reflection is a two point boundary value problem.
P
from 0 to 10 cm−1. The associated saturation intensity It can be solvedby a shooting method. The first bound-
is I =hcΓ/(2λ σ ) 500 GW/m2. ary condition is that there is no incoming field from the
sat,S S S
≈
8
z > 0. This can be written by the following relation be-
TABLE I. Photophysical parameters of the six-level system
tweentheforwardandbackwardpropagatingfieldvalues
at the back mirror sketchedinFig. 1a). Thetransitionrateskij areobtainedby
averaging data given in Ref. [32]. 1/Γ is the lifetime of level
b (L)=ρ f (L)e−2jβiL. (D3) |5i. γIR isthespectralwidthofthe1042nmzero-phononline
i back,i i at room temperature.
Fromthisstartingvalueswecandeducethevaluesofthe Parameter Value Reference
envelope amplitudes at the input mirror by integrating λP 532 nm [19]
the following differential coupled equations λS 1042 nm [19]
df 1 σP 3×10−21 m2 [33]
dzP =−2{σP[n1(z)+n2(z)]+αP}fP(z) k31 =k42 (66±5) µs−1 [32]
db 1 k35 (7.9±4.1) µs−1 [32]
P
 dddfzzS ==−2{21σ{Pσ[Sn[1n(6z()z+)−n2n(5z()z])+]+αPα}S}bPf(Sz()z) (D4) kkk466512 ((10(..5073±±±007..)85))µµµs−ss−−111 [[[333222]]]
 ddbzS = 12{σS[n6(z)−−n5(z)]+αS}bS(z), γΓIR ≈2π1×ns4−1THz [18]
where the values of the NV center density are deduced
2
from Eq. (A1). We can obtain the input I = E
0,i | 0,i| TABLE II. Physical parameters used in thesingle-pass NV−
and output Iout,i = Eout,i 2 intensities from center IR absorption measurements19 at room temperature.
| |
Optical losses are estimated from the transmission spectrum
1
E = [f (0) ρ b (0)] given in Ref. [34].
0,i i in,i i
jκin,i − (D5) Parameter Value Reference

Eout,i =ρin,iE0,i+jκin,ibi(0), n 28×1023 m−3 [19]
∗
where κ for i P,S (κ2 +ρ2 = 1) are the am- T2 150 ns [19]
plitude min,iirror tra∈ns{miss}ion cino,eifficieinn,tis. The calculation T1 2.9 ms [35] (T=300 K)
method consists in numerically optimizing the values of I0,P 400 MW/m2 [19]
fi(L) to obtain the target values of I0,i. The value of I0,S 10 MW/m2
R = I /I is then deduced with and without the
S out,S 0,S PS 16 mW
microwave field applied. This is used to calculate the
L 300 µm [19]
contrast usingEq. (8)andtheeffectivedetectedpower
max[R (CΩ ),R (0)] P . Finally, the minimum de- ΩR 2π×1.5 MHz [19]
S R S S
×
tectable magnetic field δB is evaluated using Eq. (9). C 0.003 [19] (T=300 K)
αS 0.1−0.5 cm−1 [34]
Appendix E: Tables
tectable magnetic field
The large value of T1 shows that spin relaxation is 2
negligible. Thus this is not taken into account in the δB = (F1)
rate-equation modeling of the six-level system. γC Nph(T2∗)2tm
q
where N = P λ /(hc) is the number of detected IR
ph S S
photons per second. With N the number ofIR photons
Appendix F: Sensitivity fundamental limit S
∗
collected per T we have
2
In this Appendix we derive the fundamental limit of 2
δB = . (F2)
the minimal detectable magnetic field value for methods γ N T∗t
basedonIRabsorptionorvisiblefluorescencemonitoring C S 2 m
considering that the methods are limited by the photon Now we estimate the maxipmal NS value. Assuming an
shot-noise. optimal contrast =1. When microwavesare switched-
C −
on, every photon is absorbed. We assume that one NV
∗
center absorbs M IR photons per T . In many high-
S 2
1. IR absorption based magnetometer density samples, T∗ . 1/(k +k ), and therefore we
2 61 62
∗
can consider that M <ΓT . We can thus write
S 2
Using the expression of the ESR FWHM and Eq. (2)
we obtain the following relation for the minimal de- NS =MS Nosning−Nosffing , (F3)
(cid:16) (cid:17)
9
where Nsing is the number of NV− centers in the singlet proportional to P = k /(k + k ) the probability
on 31 31 31 35
state when the microwaveareswitched-onandNsing the that NV− centers in level 3 decay immediately to level
number of NV− centers in the singlet for switcohffed-off 1 . When the microwave|s iare switched-on the fluores-
| i
microwaves. This gives the number of photons which cence signal is proportional to P42/4 + 3P31/4 where
−
can be detected when the microwaves are switched-off P42 = k42/(k42+k45) is the probability that NV cen-
tersinlevel 4 decaytolevel 2 . AssumingthatP 1
31
3 1 (k k ),|tihe contrast is| giiven by ≈
N =M N P + P P , (F4) 35 31 f
S S 35 45 35 ≪ C
4 × 4 × −
(cid:20)(cid:18) (cid:19) (cid:21)
1 k k
31 42
kw3h5e/r(ek3N5 +=k3n1)VbeisintghethneupmrobbearboilfitcyenthteartsNwVit−hcPe3n5ter=s Cf = 4(cid:18)k31+k35 − k42+k45(cid:19). (F7)
ikn l/e(vkel |+3ik(m)sth=e p0r)obdaebcialiytyttohathteNsVin−glceetntaenrds inP4l5ev=el The number of collected photons per T2∗ is Nf = ηNMf
45 45 42 where η is the collection efficiency and M the number
t|4oit(amkes i=nt±o1a)ccdoeucnatytthoatthoenslyingolneet.qTuahrete41r aonfdth34e aNllVow− of emitted photons per T2∗ by one NV− cfenter. Since
∗
1/k <T we have M <k /k .
centers are resonant with the microwaves24. We then 35 2 f 31 35
have N = M N with
S S S
R
1 k k
45 35 3. Comparison
= , (F5)
S
R 4 k +k − k +k
(cid:18) 45 42 35 31(cid:19)
The two techniques can be compared by calculating
which is an approximated value for R (Ω ) defined
S R
in section IIIB. Note that if the IR power is such as
RSMS ≥1 the sensitivity is limited by the spin-noise. δBf 1 RSMS, (F8)
δB ≈ fs ηMf
C
where we assume 1. For k k and k k ,
35 31 42 45
2. Fluorescence measurement based magnetometer C ≈ ≪ ≈
we have and thus
S f
R ≈C
Foramagnetometerusingthefluorescencesignalmon-
δB M 1
∗ f S
itoring and assuming that the ESR FWHM is 2/T , the . (F9)
sensitivity is given by22,23 2 δB ≈sMf · ηRS
2 NotethatwithvaluesrecalledinTab. II,weobtain
δBf = γ N T∗t , (F6) 8.5% which corresponds to the optimal case asumRinSg≈a
Cf f 2 m total spin polarization. We deduce that M 11 and
S
where is the contrast ofpthe fluorescence signal and M 8. Assuming that M = M and cons≤idering a
f f S f
C ∗ ≤
N the number of collected photons per T . When the high value of the collection efficiceny (η 0.47 has been
f 2 ≈
microwaves are switched-off, the fluorescence signal is reported in Ref. [15]) we obtain δB /δB 5.
f
≈
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